d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
available at www.sciencedirect.com
journal homepage: www.intl.elsevierhealth.com/journals/dema
Efficient 3D finite element analysis of dental
restorative procedures using micro-CT data
Pascal Magne ∗
University of Southern California, Division of Primary Oral Health Care, School of Dentistry, 925 West 34th Street, DEN 4366,
Los Angeles, CA 90089-0641, United States
a r t i c l e
i n f o
a b s t r a c t
Article history:
Objectives. This investigation describes a rapid method for the generation of finite element
Received 3 November 2005
models of dental structures and restorations.
Received in revised form 23 March
Methods. An intact mandibular molar was digitized with a micro-CT scanner. Surface con-
2006
tours of enamel and dentin were fitted following tooth segmentation based on pixel density
Accepted 27 March 2006
using an interactive medical image control system. Stereolithography (STL) files of enamel
and dentin surfaces were then remeshed to reduce mesh density and imported in a rapid
prototyping software, where Boolean operations were used to assure the interfacial mesh
Keywords:
congruence (dentinoenamel junction) and simulate different cavity preparations (MO/MOD
Finite element analysis
preparations, endodontic access) and restorations (feldspathic porcelain and composite
Restorative dentistry
resin inlays). The different tooth parts were then imported in a finite element software
Cuspal flexure
package to create 3D solid models. The potential use of the model was demonstrated using
Composite resins
nonlinear contact analysis to simulate occlusal loading. Cuspal deformation was measured
Porcelain inlays
at different restorative steps and correlated with existing experimental data for model validation and optimization.
Results. Five different models were validated by existing experimental data. Cuspal widening
(between mesial cusps) at 100 N load ranged from 0.4 ␮m for the unrestored tooth, 9–12 ␮m
for MO, MOD cavities, to 12–21 ␮m for endodontic access cavities. Placement of an MOD
adhesive restoration in porcelain resulted in 100% cuspal stiffness recovery (0.4 ␮m of cuspal
widening at 100 N) while the composite resin inlay allowed for a partial recuperation of cusp
stabilization (1.3 ␮m of cuspal widening at 100 N).
Significance. The described method can generate detailed and valid three dimensional finite
element models of a molar tooth with different cavities and restorative materials. This
method is rapid and can readily be used for other medical (and dental) applications.
© 2006 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
It is a well-established claim that mechanical testing is of
paramount importance, not only in aerospace, civil engineering and the automotive industry, but also in health care. The
field of biomedical research raises specific problems due to the
fact that today’s research may prove extremely expensive and
∗
ethically questionable when performed on live subjects. To
limit the costs and risks involved in live experiments, virtual
models and simulation approaches have become unavoidable
[1]: an iterative optimization of the design of the experiment
is performed on the computer and is seen in virtual prototyping and virtual testing and evaluation; after this iterative step,
when the best design has been refined, the actual experiment
Tel.: +213 740 4239; fax: +213 740 6778.
E-mail address: [email protected].
0109-5641/$ – see front matter © 2006 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.dental.2006.03.013
DENTAL-958;
No. of Pages 10
2
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
is conducted. The value is that the modeling and simulation
step saves time and money for conducting the live experiment
or clinical trial.
Yet dental research seems to make very little use of virtual models, such approaches representing a minor part of the
scientific publication volume. In finite element (FE) analysis,
a large structure is divided into a number of small simpleshaped elements, for which individual deformation (strain
and stress) can be more easily calculated than for the whole
undivided large structure. By solving the deformation of all
the small elements simultaneously, the deformation of the
structure as a whole can be assessed. Using the traditional
biophysical knowledge database in a rational validation process [2], the use of FE analysis in dental research has been
significantly refined during the last decade [3–10]. Nowadays,
experimental–numerical approaches undoubtedly represent
the most comprehensive in vitro investigation methods in
restorative dentistry [9,10]. They allow the researcher (1) to
reduce the time and cost required to bring a new idea from
concept to clinical application, (2) to increase their confidence
in the final concept/project by virtually testing it under all conceivable loading conditions.
Because teeth and bones cannot be assimilated to a simplified geometric representation but have anatomical shapes and
a layered structure, sophisticated techniques have been developed to refine geometry acquisition, such as the recreation and
digitization of planar outlines of the spatial anatomy [11,12].
This is often the most time-consuming step for the modeler.
In addition, this process is prone to errors and simplifications
which may induce faulty predictions. For this reason, patient’s
geometry-based meshing algorithms have already been proposed to generate complex solid models of bones as for example the CT scan-based FE model [6]. Similar approaches can
be used with microscale CT scanner for the simulation of
small objects like teeth, dental implants and dental restorations [13]. However, considerable work is still required in order
to obtain congruent parts (sharing the exact same geometry
at their interface) and smooth relationships between the different 3D objects (enamel, dentin, restoration). By the same
token, modification of a given parameter, like for instance
variations in restoration size, often requires the realization
of a new and separate model, including the time-consuming
geometry acquisition.
The aim of the present study is therefore to propose a further development to facilitate and accelerate geometry acquisition/modification during the fabrication of FE models of
tooth restorations. The presented method is based on stereolithography (STL) and surface-driven automatic meshing. In
this innovative approach, validated by cuspal flexure measurements, the model is built in multi-parts (using segmentation
and Boolean operations with CAD objects) based on the geometry of the unaltered tooth. The same method can also be
used to create patient-specific models from any other body
part using either MRI or CT data.
Table 1 – Sequential procedure, computer software and times used to generate the 3D FE tooth model
The tooth model may be accomplished by a trained operator in less than a workday (depending of the complexity of the parts and general goal
of the project). The same approach is applicable to other disciplines (orthodontics, orthopedic surgery, etc.) to generate patient-specific models
from MRI or CT-scan data.
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
2.
Materials and methods
2.1.
Mesh generation and material properties
(pre-processing)
A 4-step procedure (Table 1) was followed to generate a 3D FE
model of an extracted human mandibular molar.
First, the tooth was scanned with Skyscan 1072 high resolution Micro-CT (Skyscan, Aartselaar, Belgium) with a voxel
3
dimension of 13.65 ␮m. Exposure time was 7.2 s per frame, two
frames were taken per angle and there were 208◦ . A total of
1128 slices were taken in 2 h. Only 81 slices (one slice out of
every 14 slices) were used for the modeling.
Second, the different hard tissues visible on the scans
were identified using an interactive medical image control
system (MIMICS 9.0, Materialise, Leuven, Belgium). MIMICS
imports CT and MRI data in a wide variety of formats and
allows extended visualization and segmentation functions
based on image density thresholding (Fig. 1A). 3D objects
Fig. 1 – (A) CT-scan data as seen in MIMICS 9.0. The tooth is presented in three different cross-sectional views. Masks have
been applied to enamel (white) and dentin (yellow) according to voxel density thresholding. (B) 3D representation of dentin
as a result of segmentation in MIMICS.
27223
129374
a
CER
CPR
Stone base excluded.
Restoration size similar to CAV
27223
129374
23432
102612
Unrestored tooth with MO preparation
and endodontic access preparation
Unrestored tooth with MOD preparation
and endodontic access preparation
Tooth restored with MOD composite
restoration
Tooth restored with MOD feldspathic
ceramic restoration
ENDO MO
ENDO MOD
Unrestored tooth with MOD preparation
CAV MOD
Endodontic access cavity removing intact walls with both adjacent
boxes
Restoration size similar to CAV
25554
113603
21510
95826
27223
23519
129374
106817
Mandibular molar
Occlusal width: 1/2 intercusp. width; occlusal depth: >3 mm;
proximal depth: 1 mm above CEJ; distal marginal ridge intact
Occlusal width: 1/2 intercusp. width; occlusal depth: >3 mm;
proximal depth: 1 mm above CEJ
Endodontic access cavity removing intact wall with adjacent box.
Intact natural tooth (unrestored)
Unrestored tooth with MO preparation
NAT
CAV MO
Description
Model label
Fig. 2 – (A) Stereolithography triangulated (STL) file of
enamel obtained through the STL+ module within MIMICS.
The density and quality (aspect ratio and connectivity) of
the triangles is not appropriate for use in finite element
analysis. (B) Enamel STL file optimized for FEA using the
REMESH module within MIMICS. Note the improved
triangle shape and the intact geometry compared to Fig. 2A
in spite of a significant reduction in number of triangles.
Table 2 – FEA geometry and characteristics for the different models
Specific features
Volumetric mesh
(enamel and dentin) are automatically created in the form of
masks by growing a threshold region on the entire stack of
scans (Fig. 1B). Using MIMICS STL+ module, enamel (Fig. 2A)
and dentin were then separately converted into stereolithography files (STL, bilinear and interplane interpolation algorithm). Native STLs are improper for use in FEA because of
the aspect ratio and connectivity of the triangles in these files.
The REMESH module attached to MIMICS was therefore used
to automatically reduce the amount of triangles and simultaneously improve the quality of the triangles while maintaining
the geometry (Fig. 2B). During remesh, the tolerance variation
from the original data can be specified (quality of triangles
does not mean tolerance variation from the original data). The
quality is defined as a measure of triangle height/base ratio
so that the file can be imported in the finite element anal-
No. of nodesa
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
No. of elementsa
4
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
ysis software package without generating any problem. The
remesh operations were also applied to the dentin STL. Segmentation of enamel and dentin may be accomplished by a
trained operator in ca. 90 min (including remesh of the STL
files).
Third, a strereolithography handling software (MAGICS 9.9,
Materialise, Leuven Belgium) was used in order to reestablish the congruence of the interfacial mesh between enamel
and dentin (this congruence being lost during the previous
remeshing process) using Boolean operations (addition, inter-
5
section or subtraction of volumes). Once a congruent mesh at
the dentinoenamel junction was obtained (Fig. 3A), additional
Boolean operations with CAD objects (Fig. 3B and C) were used
to simulate a cylindrical fixation base (embedding the root
within 2 mm of the cementoenamel junction), as well as different cavity preparations (MO and MOD cavities, endodontic
access cavities) and restorations. The exact design and dimensions of the MO, MOD and endodontic access cavities are
described in Table 2. These successive restorative situations
were chosen because they reproduce existing experiments
Fig. 3 – (A) Cross-sectional view of the enamel–dentin assembly as seen in MAGICS. Both enamel and dentin STLs share the
exact same geometry at their interface (dentinoenamel junction). (B). Congruent enamel (white) and dentin (yellow) meshes
along with CAD objects used to simulate a stone base (cylinder) and different cavity designs (red inserts). Mesh congruence
between the root portion and the stone base was obtained through a Boolean subtraction process (stone cylinder minus
dentin). (C). Congruent STL parts of enamel (white) and dentin (yellow) resulting from Boolean intersections and
substractions between the original enamel/dentin STLs and different CAD inserts (right side). The assembly of the different
parts result in five possible models (left side), i.e. the natural tooth (NAT), MO and MOD cavities (CAV), MO and MOD
endodontic access preparations (ENDO). The NAT model was also used to simulate a composite resin inlay (CPR) and a
porcelain inlay (CER) by attributing different material properties to the enamel and dentin inserts (see Table 2).
6
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
Table 3 – Material properties
Elastic modulus (GPa)
Enamel
Dentin
Composite
Ceramic
a
b
c
d
e
f
g
Poisson’s ratio
84.1a
18.6c
10.0e
78.0g
0.30b
0.31d
0.24f
0.28a
Craig et al. [25].
Anusavice and Hojjatie [24].
McGuiness et al. [28].
Farah et al. [27].
Eldiwany et al. [26].
Nakayama et al. [29].
Data from manuftacturer of Creation-Willi Geller dental porcelain
(Klema, Meiningen, Austria).
[14–16], which will be used in the validation process of the FEA
model (see Section 3). Two additional experimental conditions
were generated by attributing different material properties to
the enamel and dentin inserts included in model NAT: a composite resin inlay (CPR) and a feldspathic ceramic inlay (CER).
The treatment of the STL files in MAGICS may be accomplished
by a trained operator in ca. 30–60 min per model (including all
Boolean operations).
Fourth, the optimized STL files of the segmented enamel
and dentin parts were then imported in a finite element analysis software package (MSC.Marc/MSC.Mentat, MSC.Software,
Santa Ana, CA) for the generation of a volumetric mesh and
attribution of material properties (Table 3). The triangulated
STL files are ideal for automatic mesh generation using a
tetrahedral mesher (tetrahedron elements with pyramid-like
shape and 4 nodal points). This last step may be accomplished
by a trained operator in ca. 30–60 min per model (including
attribution of boundary conditions and 17 min to run the analysis).
Fig. 4 – Load protocol and configuration as seen in Mentat,
i.e. a nonlinear contact analysis between a rigid body
(9.5-mm diameter load sphere moving along Z-axis against
the tooth) and a deformable tooth (CAV MOD shown here).
The widening of the cusps (v) was calculated from the
output values of displacement along the Y-axis for selected
nodes near the cusp tip.
2.2.
Boundary conditions, loadcase and data
processing
3.
Fixed zero-displacement in the three spatial dimensions was
assigned to the nodes at the bottom surface of the stone base.
The tooth and restorative materials were taken as bonded,
which simulate usage of adhesive luting cements. A uniformly
ramp loading was applied to the mesial cusps through a rigid
body, i.e. a 9.5-mm diameter ball positioned as close as possible to the tooth (Fig. 4). The tooth was defined as deformable
contact body. Contact between these bodies was determined
automatically by the FEA simulation during the static mechanical loadcase (no inertia effects) with a uniform stepping procedure of 10 steps. A motion was applied to the rigid ball along
the Z-axis through a negative velocity of 0.02 mm per step.
Only one step was required to reach contact in both cusps. The
motion continued for the remaining steps to reach a total force
100–200 N on the ball (depending on the model). The stress and
strain distributions were solved using the MSC.Marc solver. As
mentioned before, these specific boundary conditions, load
protocol and configuration were chosen because they reproduce existing experiments by Panitvisai and Messer [14], and
Jantarat et al. [15,16].
Results
The post-processing file was accessed through MENTAT to
select specific nodes on the buccal and lingual enamel near
the cusp tip and to collect the values of displacement in
the Y direction for each loading step (Y+ denotes displacement in lingual direction and Y− in buccal direction). The
force along the Z-axis on the rigid ball was also collected
for each step. After the transfer of these data to a spreadsheet, the widening (deformation) of the cusp was calculated
(by summing the displacement of each cusp) and plotted
against the force along Z-axis on the ball (Fig. 5). As expected
in an elastic simulation, there is a quasi-linear relationship
between load and deformation. The progressive loss of tooth
substance (MO to MOD to ENDO) translates into a progressive loss of cuspal stiffness (decreased slope of the force vs.
deformation plot). The unrestored tooth (NAT) and the tooth
with the MOD ceramic inlay (CER) display the same conduct (100% recovery of cuspal stiffness), while the more flexible composite inlay allowed for partial of recovery of cuspal
stiffness.
7
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4.
Fig. 5 – Force generated by the load ball in Newtons vs.
cuspal widening in millimeters (mesial cusps) for each
experimental design.
FEA validation
For validation of the models, the widening of the mesial cusp
at a load of 100 N for all groups was extracted and is presented
in Table 4 along with results from existing experiments by
Panitvisai and Messer [14] and Jantarat et al. [15,16] who used
the same type of tooth, loading protocol (static load at 100 N)
and configuration (9.5 mm load sphere seating on the mesial
cusps).
There is a good association between FEA and experimental values at each restorative step, the difference being inferior to 3 ␮m in most conditions. A larger difference was
noted when comparing the MOD endodontic accesses in the
FEA to the study by Panitvisai and Messer: [14] 21.3 ␮m versus 28.8 ␮m respectively. This discrepancy, however, does not
appear when comparing the same model with the studies
by Jantarat et al. [15,16] The MOD composite yielded 1.3 ␮m
of cuspal widening (3× the value of NAT), unlike the MOD
ceramic restoration, which was similar to the unaltered tooth
(0.4 ␮m).
Discussion
A number of studies [17–20] analyzing biophysical stress and
strain have shown that restorative procedures can make the
tooth crown more deformable, and teeth could be strengthened by increasing their resistance to crown deformation. The
standard loadcase applied in the present analysis constitutes
the most discriminating technique to study crown deformation; it also constitutes a useful validation set-up that mirrors
existing experimental cuspal flexure measurements. Jantarat
et al. [16] used an extensometer and only measured teeth with
an opened MOD endodontic access. Panitvisai and Messer
[14] and Janatarat et al. [15] used direct current displacement
transducers (DCDT) to measure individual cusp displacement.
The precision of each transducer being around 1 ␮m, they were
not able to measure submicron displacements. In addition, a
cumulated error of 2 ␮m can be expected on the total cusp
widening. The maximum difference found between the experimental measurements and the FEA model was around 3 ␮m.
Considering that the inter-tooth variability reported by both
articles far exceeds these values, the model can be considered valid. The results obtained with composite restored teeth
(CPR) are in agreement with conclusions by Douglas [17] stating that their strength falls off with increasing cavity size and
can only approach that of the unaltered tooth in the case of
small conservative cavities. This cannot be said about ceramic
restored teeth (CER), the behavior of which is strictly mimicking the unaltered tooth (Fig. 6). These results are in agreement
with in vivo and in vitro studies [21,22] showing tooth-like
fracture resistance, better cuspal protection and significantly
better “anatomical form of the surface” and “integrity of the
restoration” for ceramic inlays compared to composite ones.
As illustrated in the present article, the proposed approach
resulted in valid 3D models with very detailed tooth anatomy
and realistic computation process. Previous attempt to generate 3D models resulted in much coarser meshes [11,12], mainly
due to the limitation of the geometry acquisition method
(manual tracing of actual tooth sections), another reason being
the increased memory requirements for 3D models, which did
not allow fine representation of the geometry. Other authors
[3–5] digitized a plaster model (crown portion) and extrapolated the inner geometry (pulp, root dentin and enamel
volumes) using tooth morphology literature data. Different
Table 4 – Results and comparison with existing experimental data
Widening v ( ␮m) at 100 N load (force Z)
Experimental condition
FEA
NAT (intact tooth)
CAV MO (MO cavity)
CAV MOD (MOD cavity)
ENDO MO (MO + endo. access)
ENDO MOD (MOD + endo. access)
CER (MOD ceramic inlay)
CPR (MOD composite inlay)
0.4
9.1
11.8
12.3
21.3
0.4
1.3
Panitvisai and Messer [14]
<1
6
10
14.4
28.8
–
–
Jantarat et al.
[15]
[16]
<2
–
6–10
–
24–32
–
–
–
–
–
–
17–18
–
–
8
d e n t a l m a t e r i a l s x x x ( 2 0 0 6 ) xxx–xxx
Fig. 6 – First principal stress distribution in four of the seven models studied. To allow for better comparison of the stress
pattern, the MOD insert of enamel and dentin (model NAT) or the MOD ceramic restoration (model CER) were made invisible.
Colors, tensile stresses; gray, compressive stresses. Note the similarity between NAT and CER.
approaches were proposed to access the inner anatomical
detail without extrapolation and accelerate the production
of the models. Verdonschot et al. [13] might have been the
first authors to describe the development of a 3D finite element model of a restored tooth based on a microscale CT
data-acquisition technique. The tooth was scanned after being
restored with an MOD composite and the 3D geometry was
obtained through the stacking of traced 2D sections, still
involving a significant amount of manual work. An interesting semi-automated method was proposed [6–8] to generate
solid models of bones without internal boundaries (plain automatic volumetric mesh), then using the Hounsfield unit (HU)
to attribute a specific Young’s modulus to each element based
on scan density. When applied to small structures like teeth
(with thin anatomical details such as the enamel shell), this
technique does not allow the fine control of internal boundaries (e.g. dentinoenamel junction), the exact geometry of
which will have to follow the automatic volumetric meshing
process.
The approach used in the present study suggests that
maximum anatomical detail is obtained by surface/interfacebased meshing using stereolithography (STL) surface data.
The different parts of the model featuring different mechanical properties are identified first (segmentation process) and
meshed accordingly. Elements do not overlap the different
structures but strictly follow the internal boundaries, resulting
in a smooth and very well controlled representation of inter-
faces like the dentinoenamel junction (Fig. 3A). Significant
advantages, when using STLs, are the sophisticated visualization tools (shaded wireframe 3D views, section views etc.)
and possibilities offered by the Boolean operations. The general principle of Boolean operations is that a new object can be
formed by combining two 3D objects. Objects can be united,
intersected or subtracted. When intersecting or subtracting
two overlapping objects, a congruent mesh is assured at the
interface between the new objects. This property is essential to assure the continuity of the resulting volumetric mesh.
Boolean operations with predefined CAD objects (box, cylinder, cone or inserts as in Fig. 3B) constitute an important
feature. It allowed us to “digitally” simulate successive restorative procedures (Fig. 3C), unlike Verdonschot et al. [13] who
had to “physically” restore the tooth before scanning it. In the
present study, the geometry of the unaltered tooth remains,
allowing for direct comparison with the different experimental conditions. The very user-friendly graphic interface allows
for rapid modifications of the different parts and generations
of new STLs that can be instantly exported and volumetrically
meshed the FEA program. It must be pointed out that micro-CT
is not suitable for human teeth in live patients. However, considering that only 81 slices were necessary to generate these
valid FEA models, one can easily foresee that the exponential
development of commercial dental CT-scanners, computer
processing power and interface friendliness will make this
approach even faster and more automated, allowing the rapid
dental materials
fabrication of patient-specific simulation of dental restorations in a very near future. Even though small differences
may remain between the reality and the finite element environment, numerical modeling is able to reveal the otherwise
inaccessible stress distribution within the tooth-restoration
complex (Fig. 6) and it has proven to be a useful tool in the
thinking process for the understanding of tooth biomechanics and the biomimetic approach in restorative dentistry [23].
5.
Conclusion
This investigation describes a rapid method for the generation
of finite element models of dental structures and restorations.
Detailed three dimensional finite element models of a molar
tooth with different cavities and restorative materials were
generated. The potential use of the model was demonstrated
using nonlinear contact analysis to simulate occlusal loading.
Cuspal widening was measured at different restorative steps
and correlated with existing experimental data for model validation and optimization. This method is rapid (a tooth model
may be obtained by a skilled operator in less than a workday)
and can readily be used for other medical applications to create patient-specific models from any other body part using
either MRI or CT data. Further, this methodology could facilitate optimization and understanding of biomedical devices
prior to animal and human clinical trials.
Acknowledgements
The author wish to express his gratitude to Tim Sledz (Micro
Photonics Inc., distributor of Skyscan in the USA) for scanning
the experimental sample. This study was supported in part by
MSC.Software (MSC.Marc/MSC.Mentat products) and Materialise (MIMICS/MAGICS products). Special thanks to Dan Wolf
(MSC.Software) for helpful suggestions and Mrs. Constance
Nelson who has given in memory of Dr. Gus Swab.
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